Pragmatic, Unifying Algorithm Gives Power Probabilities for Common F Tests of the Multivariate General Linear Hypothesis

We consider the problem of computing the power of some usual F transforms of the Wilks U, Hotelling-Lawley T, and Pillai V statistics for testing H0: CBA = Œ0 under the multivariate general linear model, Y = XB + ‰ , where the rows of ‰A are taken as independent N(0, Í) random vectors. Keeping all these matrices at full rank, let C be rC × rX and A be P × rA. For determining p-values, Fi (i ∈ {U, T1, T2, V}) is taken to be distributed as central F(rCrA, ν (i) 2 ), which is the exact distribution when s = min(rC, rA) = 1. For determining powers, we present a pragmatic, unifying method that takes Fi to be noncentral F(rCrA, ν (i) 2 , λi), where λi is isomorphic to Fi. For any s, we obtain the simple form λi = Nλi , where λi is not a function of the total sample size, N. We show that for s = 1, F(rCrA, ν (i) 2 , λi) defines the exact noncentral distribution. For s > 1, each Fi converges in distribution to its prescribed noncentral F distribution and numerical work supports the accuracy of all approximations for obtaining powers for all but very small N. We exploit the method to compare the powers of the various Fi statistics. Finally, we illustrate the method by computing a set of powers for a multivariate analysis of variance comparing the profiles of three correlated tests among three independent groups.